A group-theoretic approach to fast matrix multiplication

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n^(2 + o(1)) support n-by-n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page numbers and copyright informatio

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur